
where \({\mathcal {S}}^n_+\)  is the positive semidefinite matrix cone, \({\mathcal {N}}_n\)  is the cone of nonnegative matrices and \(\mathcal {DNN}\)  abbreviates doubly-non-negative. In fact, the inclusion in (REF ) holds with equality in case \(n<5\)  (see [1]}). More elaborate approximations based on SDPs or LPs are listed in [2]}. For approximation hierarchies for a more general class of cones \({\mathcal {K}}\)  see [3]}. SDPs are amenable to interior point methods and are thus solvable in polynomial time. But in practice these methods for SDPs are known scale poorly and are not as numerically stable as, say, modern MILP-sovlers. Solving large scale SDPs requires substantial effort and is only possible for problems with exploitable special structure. On typical office-level computers only medium sized general SDPs can be solved. There are LP approximations for \(\mathcal {CPP}(\mathbb {R}^n_+)\)  but the advantage over SDPs is diminished and might even be outweighed by the size of these approximations. Recent literature (see [4]}) introduced cutting plane algorithms for copositive optimization problems, where the subproblem of copositivity detection is solved via MILPs. However, these methods have not yet shown that they can remedy the computational expensiveness of completely positive optimization problems. This shortcoming motivates the search for smaller and cheaper convex approximations and reformulations of QCQPs, which may be achieved by exploiting special structure. In this spirit, the main goal of this article is to provide a reduced convex relaxation/reformulation of a special class of QCQPs, whose special structure naturally arises in e.g. stochastic optimization. Compared to traditional approaches, our relaxations/reformulations will omit spurious terms and involve conic constraints of lower dimension.
